High School

We appreciate your visit to An airplane manufacturer claims that the average weight of their jet engines is 139 302 kg An auditor inspects 30 randomly selected engines and finds. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

An airplane manufacturer claims that the average weight of their jet engines is 139.302 kg. An auditor inspects 30 randomly selected engines and finds that their average weight is 139.3 kg. Given a standard deviation of 3.02 kg:

(a) What is the probability that a single engine weighs more than the stated average of 137 kg? (Round your answer to four decimal places.)

(b) Among ten randomly selected engines, what is the probability that at least one engine weighs more than the stated average? (Round your answer to four decimal places.)

(c) Assuming that the mean remains at 139.3 kg, to what value would the standard deviation have to be changed so that 95% of all engines weigh more than the stated average of 137 kg? (Round your answer to two decimal places.)

Answer :

(a) The probability that a single jar contains more than the stated contents is approximately 0.1587.

(b) The probability that at least one jar among ten randomly selected jars contains more than the stated contents is approximately 0.8312.

(c) To ensure that 95% of all jars contain more than the stated contents with a mean of 139.3, the standard deviation would need to be changed to approximately 1.79.

(a) To find the probability that a single jar contains more than the stated contents, we can use the standard normal distribution. First, we calculate the z-score:

[tex]\[ z = \frac{137 - 139.3}{3.02} \approx -0.76159 \][/tex]

Using a standard normal distribution table or calculator, we find that the probability corresponding to this z-score is approximately 0.7787. Since we are interested in the probability that a jar contains more than the stated contents, we subtract this value from 1:

[tex]\[ P(X > 137) = 1 - 0.7787 \approx 0.2213 \][/tex]

(b) To find the probability that at least one jar among ten randomly selected jars contains more than the stated contents, we can use the complement rule. First, we find the probability that none of the jars contain more than the stated contents:

[tex]\[ P(\text{none contain more than stated contents}) = (1 - 0.2213)^{10} \approx 0.1688 \][/tex]

Then, the probability that at least one jar contains more than the stated contents is:

[tex]\[ P(\text{at least one contains more than stated contents[/tex][tex]}) = 1 - 0.1688 \approx 0.8312 \][/tex]

(c) To find the value of the standard deviation that would ensure that 95% of all jars contain more than the stated contents with a mean of 139.3, we can use the z-score formula for a given percentile in the standard normal distribution:

[tex]\[ z = \Phi^{-1}(0.95) \approx 1.645 \][/tex]

Then, we solve for the standard deviation:

[tex]\[ 1.645 = \frac{137 - 139.3}{\text{standard deviation}} \][/tex]

[tex]\[ \text{standard deviation} = \frac{139.3 - 137}{1.645} \approx 1.79 \][/tex]

Thanks for taking the time to read An airplane manufacturer claims that the average weight of their jet engines is 139 302 kg An auditor inspects 30 randomly selected engines and finds. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada